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Properties

Label 20.20.5932229289...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{8}\cdot 5^{26}\cdot 41^{15}$
Root discriminant $173.25$
Ramified primes $2, 5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_5:D_5.Q_8$ (as 20T105)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-672164, -21836260, 322525720, 1254927950, 714683790, -1902871615, -2366271710, -68831430, 843001245, 158724730, -131147817, -28324355, 10985540, 2177710, -517835, -82815, 13450, 1495, -180, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 180*x^18 + 1495*x^17 + 13450*x^16 - 82815*x^15 - 517835*x^14 + 2177710*x^13 + 10985540*x^12 - 28324355*x^11 - 131147817*x^10 + 158724730*x^9 + 843001245*x^8 - 68831430*x^7 - 2366271710*x^6 - 1902871615*x^5 + 714683790*x^4 + 1254927950*x^3 + 322525720*x^2 - 21836260*x - 672164)
 
gp: K = bnfinit(x^20 - 10*x^19 - 180*x^18 + 1495*x^17 + 13450*x^16 - 82815*x^15 - 517835*x^14 + 2177710*x^13 + 10985540*x^12 - 28324355*x^11 - 131147817*x^10 + 158724730*x^9 + 843001245*x^8 - 68831430*x^7 - 2366271710*x^6 - 1902871615*x^5 + 714683790*x^4 + 1254927950*x^3 + 322525720*x^2 - 21836260*x - 672164, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 180 x^{18} + 1495 x^{17} + 13450 x^{16} - 82815 x^{15} - 517835 x^{14} + 2177710 x^{13} + 10985540 x^{12} - 28324355 x^{11} - 131147817 x^{10} + 158724730 x^{9} + 843001245 x^{8} - 68831430 x^{7} - 2366271710 x^{6} - 1902871615 x^{5} + 714683790 x^{4} + 1254927950 x^{3} + 322525720 x^{2} - 21836260 x - 672164 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(593222928997626461368027114868164062500000000=2^{8}\cdot 5^{26}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $173.25$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41} a^{12} - \frac{16}{41} a^{11} - \frac{7}{41} a^{10} + \frac{12}{41} a^{9} - \frac{15}{41} a^{8} + \frac{4}{41} a^{7} + \frac{11}{41} a^{6} + \frac{17}{41} a^{5} + \frac{2}{41} a^{4} + \frac{15}{41} a^{3} - \frac{18}{41} a^{2} + \frac{9}{41} a + \frac{10}{41}$, $\frac{1}{41} a^{13} - \frac{17}{41} a^{11} - \frac{18}{41} a^{10} + \frac{13}{41} a^{9} + \frac{10}{41} a^{8} - \frac{7}{41} a^{7} - \frac{12}{41} a^{6} - \frac{13}{41} a^{5} + \frac{6}{41} a^{4} + \frac{17}{41} a^{3} + \frac{8}{41} a^{2} - \frac{10}{41} a - \frac{4}{41}$, $\frac{1}{41} a^{14} - \frac{3}{41} a^{11} + \frac{17}{41} a^{10} + \frac{9}{41} a^{9} - \frac{16}{41} a^{8} + \frac{15}{41} a^{7} + \frac{10}{41} a^{6} + \frac{8}{41} a^{5} + \frac{10}{41} a^{4} + \frac{17}{41} a^{3} + \frac{12}{41} a^{2} - \frac{15}{41} a + \frac{6}{41}$, $\frac{1}{41} a^{15} + \frac{10}{41} a^{11} - \frac{12}{41} a^{10} + \frac{20}{41} a^{9} + \frac{11}{41} a^{8} - \frac{19}{41} a^{7} + \frac{20}{41} a^{5} - \frac{18}{41} a^{4} + \frac{16}{41} a^{3} + \frac{13}{41} a^{2} - \frac{8}{41} a - \frac{11}{41}$, $\frac{1}{82} a^{16} - \frac{1}{82} a^{14} - \frac{1}{82} a^{13} - \frac{1}{82} a^{12} + \frac{10}{41} a^{11} + \frac{8}{41} a^{10} - \frac{10}{41} a^{9} - \frac{6}{41} a^{8} - \frac{11}{82} a^{7} - \frac{17}{82} a^{6} + \frac{5}{82} a^{5} - \frac{11}{41} a^{4} + \frac{19}{82} a^{3} + \frac{3}{41} a^{2} + \frac{19}{41} a - \frac{15}{41}$, $\frac{1}{82} a^{17} - \frac{1}{82} a^{15} - \frac{1}{82} a^{14} - \frac{1}{82} a^{13} + \frac{4}{41} a^{11} + \frac{19}{41} a^{10} - \frac{3}{41} a^{9} - \frac{39}{82} a^{8} - \frac{15}{82} a^{7} + \frac{31}{82} a^{6} - \frac{17}{41} a^{5} - \frac{21}{82} a^{4} + \frac{17}{41} a^{3} - \frac{6}{41} a^{2} + \frac{18}{41} a - \frac{18}{41}$, $\frac{1}{82} a^{18} - \frac{1}{82} a^{15} - \frac{1}{82} a^{13} - \frac{1}{82} a^{12} + \frac{8}{41} a^{11} + \frac{9}{41} a^{10} + \frac{27}{82} a^{9} - \frac{21}{82} a^{8} + \frac{9}{41} a^{7} - \frac{37}{82} a^{6} + \frac{14}{41} a^{5} + \frac{8}{41} a^{4} + \frac{3}{82} a^{3} - \frac{18}{41} a^{2} - \frac{9}{41} a - \frac{8}{41}$, $\frac{1}{77751291611669928934710131308686205063415215053620367977167196214312477404} a^{19} - \frac{217695929354643620994497402055975386752931010495440974719737827700448639}{77751291611669928934710131308686205063415215053620367977167196214312477404} a^{18} - \frac{80979044754618835122018311840331806364957642983938508200061038354520131}{77751291611669928934710131308686205063415215053620367977167196214312477404} a^{17} - \frac{9804341498080927530749514954567498732350503173767638832708625225316108}{19437822902917482233677532827171551265853803763405091994291799053578119351} a^{16} - \frac{209548958057010789272008719095446184750205164564597914144033071945351761}{19437822902917482233677532827171551265853803763405091994291799053578119351} a^{15} - \frac{1769562779698433239747940500634738562308666456945453598082767743935323}{1065086186461231903215207278201180891279660480186580383248865701565924348} a^{14} + \frac{204993088958812393609625408201706622971111915536635849416147926308065273}{19437822902917482233677532827171551265853803763405091994291799053578119351} a^{13} + \frac{56013045666523157646998988576716692078106948780967367423861134123097903}{19437822902917482233677532827171551265853803763405091994291799053578119351} a^{12} + \frac{44980320594110022575373221153840020700861436179015564325084938437387626}{474093241534572737406769093345647591850092774717197365714434123258002911} a^{11} - \frac{17342726044434952683400014277946317880312895196124848158738369662297277059}{77751291611669928934710131308686205063415215053620367977167196214312477404} a^{10} - \frac{5295982258068218080506507311054569073013818093176964378024163587345541849}{38875645805834964467355065654343102531707607526810183988583598107156238702} a^{9} - \frac{758473340553553281863153507932428751003863138302662345979712900727624413}{38875645805834964467355065654343102531707607526810183988583598107156238702} a^{8} - \frac{3785608889098848905921200101674375117623478229917273902282083591137814473}{77751291611669928934710131308686205063415215053620367977167196214312477404} a^{7} + \frac{27709867048622577662598280239278731744611992477001727105958142212990810007}{77751291611669928934710131308686205063415215053620367977167196214312477404} a^{6} - \frac{30033810033839153445500839012873149414167463401467507053155315304623868139}{77751291611669928934710131308686205063415215053620367977167196214312477404} a^{5} + \frac{663811805996149750888431605165560728317460916353957993801983793099082215}{38875645805834964467355065654343102531707607526810183988583598107156238702} a^{4} - \frac{439617350685009618873659068596652160413993853589464571421573962353788439}{38875645805834964467355065654343102531707607526810183988583598107156238702} a^{3} - \frac{8065441143788813198986796454070211529401019470648523908317600868100657148}{19437822902917482233677532827171551265853803763405091994291799053578119351} a^{2} - \frac{8279618795502959902783746748261781449895578982840643531530704607166887178}{19437822902917482233677532827171551265853803763405091994291799053578119351} a + \frac{645550428639193519030357442667785351367853553281758851688037475607167164}{19437822902917482233677532827171551265853803763405091994291799053578119351}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 103193338100000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_5.Q_8$ (as 20T105):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_5:D_5.Q_8$
Character table for $C_5:D_5.Q_8$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.1723025.1, 10.10.452563285156250000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R $20$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.10.15.16$x^{10} - 10 x^{6} + 10$$10$$1$$15$$F_{5}\times C_2$$[7/4]_{4}^{2}$
5.10.11.7$x^{10} + 5 x^{2} + 10$$10$$1$$11$$F_{5}\times C_2$$[5/4]_{4}^{2}$
41Data not computed